Elliptic analogue of the Hardy sums related to elliptic Bernoulli functions
Yilmaz Simsek
Abstract
In this paper, we define generalized Hardy-Berndt sums and ellip- tic analogue of the generalized Hardy-Berndt sums related to elliptic Bernoulli polynomials. We give relations between the Weierstrass
℘(z)-function, Hardy-Bernd sums, theta functions and generalized Dedekind eta function.
2000 Mathematical Subject Classification: Primary 11F20, 11B68;
Secondary 14K25, 14H42.
Key words and phrases: Bernoulli polynomials and Bernoulli functions,, Dedekind sums, Dedekind-Rademacher sums, Hardy sums, Theta functions.
1 Introduction, definitions and notations
The Dedekind eta function, η(z) is defined by η(z) = eπiz12
∞ n=1
(1−e2πinz), 3
where z ∈H = {z ∈C:Im(z)>0}. The behavior of this function under the modular group, Γ(1) is given by the following functional equation:
Theorem 1. ([1]) Let A =
⎡
⎣ a b c d
⎤
⎦∈Γ(1),
logη(Az) = logη(z) + πi(a+d)
12c −πi(s(d, c)−1 4) + 1
2log(cz+d), where s(d, c) is the Dedekind sum, which is defined as follows:
s(d, c) =
c−1
n=1
((n
c))((dn c )), and
((x)) =
⎧⎨
⎩
x−[x]− 12, x /∈Z 0, x∈Z
[x] denotes the greatest integer functions cf. (see also [1], [6], [3], [2], [7], [22], [11], [23], [25], [30]).
Generalized Dedekind eta functions defined by [22]:
Letg and h be integers, and N be a positive integer. We define ηg,h(z;N) =αg,h(N) exp(πizB2
g N
)m≡g(N), m>0
1−ζhNqNm
m≡−g(N), m>0
1−ζ−NhqmN
for z ∈H where ζN =e2πiN , qN =e2πizN and αg,h(N) =
⎧⎨
⎩
exp(πiB1h
N
)
1−ζN−h
, if g ≡0, h≡modN, 1, otherwise.
B1 and B2 in the formulae are Bernoulli functions:
B1(x) =x−[x]− 1
2, B2(x) = (x−[x])2−(x−[x]) + 1 6.
Here Bn(x) is the nth Bernoulli functions, which are defined by Bn(x) =Bn(x−[x]) =
⎧⎨
⎩
0, if n= 1, x∈Z, Bn({x}) , otherwise
, where Bn(x) denotes Bernoulli polynomials:
tetx et−1 =
∞ j=0
Bj(x)tj j!,
cf. ([34], [35], [38], [31]). The functionsηg,h(z;N) are holomorphic forz ∈H and depend upon g, h modulo N. Furthermore, ηg,h(z;N) =η−g,−h(z;N) for each g and h,and ηg,h(z;N) =η2(z) for (g, h)≡(0,0) (modN) (see for detail [22], [37], [19], [23], [25]).
Halbritter[12] and Hall et at.[13] defined generalized Dedekind sums as follows:
Definition 1. Let a, b and c be positive integers, and x, y and z be real numbers.
(1) Sm,n(a, b, c:x, y, z) =
kmodc
Bm(ak+z
c −x)Bn(bk+z c −y).
The classical theta-functions,ϑn(0, q)(n = 2, 3, 4) are defined as follows ([38], [16], [24], [30])
ϑ2(0, q) = 2q14
n=0
qn(n+1) = 2q14 ∞ n=1
(1−q2n)(1 +q2n)2, ϑ3(0, q) = 1 + 2
n=1
qn2 = ∞ n=1
(1−q2n)(1 +q2n−1)2, ϑ4(0, q) = 1 + 2
n=1
(−1)nqn2 = ∞ n=1
(1−q2n)(1−q2n−1)2,
q = eπiz, z ∈ C, |q| < 1. Throughout of this paper, we denote ϑ2(0, q), ϑ3(0, q) and ϑ4(0, q) by ϑ2(q),ϑ3(q) and ϑ4(q), respectively.
Relations between theta functions and η-function are given by (see [16], [19], [23]):
(2) ϑ2(z) = 2η2(2z)
η(z) , ϑ3(z) = η5(z)
η2(2z)η2(z2), ϑ4(z) = η2(z2) η(z) .
Letτ ∈H,e(x) = e2πix. Jacobi’s theta function is defined by ([17], [18]) θ(x, τ) =
m∈Z
e
(m+12)2
2 τ+ (m+ 1
2)(x+1 2)
.
Observe that θ(x+ 1, τ) = −θ(x, τ), θ(x+τ, τ) = −e(−τ2 −x)θ(x, τ).
In [17] and [18], Machide defined the following generating function of the elliptic Bernoulli functions (Kronecker’s double series), Bm(y, x;τ):
F(y, x;a, τ) = e(ax)f(−y+xτ, a;τ)
= ∞ m=0
Bm(y, x;τ)(2πi)m m! am−1, where
f(x, a;τ) =
∂θ(x,τ)
∂x θ(x+a, τ) θ(x, τ)θ(a, τ) , x, a ∈C\Z+τZ.
We note that Bm(y+ 1, x;τ) = Bm(y, x+ 1;τ) = Bm(y, x;τ) cf. ([17], [18], [21]).
Proposition 1. ([18]) Let x and y be real numbers and y /∈ Z. Then we have
τlim→i∞Bm(y, x;τ) =
⎧⎨
⎩
1+e(y)
2(1−e(y)) = icot(2yπ), m= 1 and x∈Z Bm(x), otherwise
,
especially
Re
τlim→i∞Bm(y, x;τ)
=Bm(x). Here Rew means the real part of a complex number w.
In [18], Machide defined the elliptic Dedekind-Rademacher sums as fol- lows:
Let a, a, b, b, c, c be positive integers and x, x, y, y, z, z be real numbers. Suppose that
az −cx ∈/< a, c >Z and bz −cy ∈/< b, c >Z, where < a, b > is the greatest common divisor ofa and b.
Set (−→a ,−→
b ,−→c) = ((a,a),(b, b),(c, c)), (−→x ,−→y ,−→z) = ((x, x),(y,y),(z,z)).
The elliptic Dedekind-Rademacher sums are defined by [18]
Sm,nτ (−→a ,−→
b ,−→c;−→x ,−→y ,−→z ) = jmodc lmodc
Bm
al+z
c −x,aj+z c −x;a
aτ (3)
×Bn
bl+z
c −y,bj+z c −y;b
bτ
. Observe that if m= 1, n = 1, or ifaz −cx /∈< a, c >Z and bz−cy /∈< b, c >Z, then
τlim→i∞Sm,nτ (−→a ,−→
b ,−→c;−→x ,−→y ,−→z ) =Sm,n(a, b, c;x, y, z).
Hardy-Berndt sums derived from the transformation formulae for logϑn(z) (n = 2, 3, 4), which are similar to Dedekind sum. Forh, k ∈Zwith k > 0,
Hardy-Berndt sums are defined as follows ([7], [23], [26], [36]):
S(h, k) =
k−1
j=1
(−1)j+1+[hjk], s1(h, k) = k j=1
(−1)[hjk]((j k)), (4)
s2(h, k) = k
j=1
(−1)j((j k))((hj
k )), s3(h, k) = k
j=1
(−1)j((hj k )), s4(h, k) =
k−1
j=1
(−1)[hjk], s5(h, k) = k j=1
(−1)j+[hjk]((j k)).
The relations between Hardy sums and Dedekind sums are given as follows:
Theorem 2. ([36]) Let (h, k) = 1. Then if h+k is odd, S(h, k) = 8s(h,2k) + 8s(2h, k)−20s(h, k), if h is even,
s1(h, k) = 2s(h, k)−4s(h,2k), if k is even,
s2(h, k) =−s(h, k) + 2s(2h, k), if k is odd,
s3(h, k) = 2s(h, k)−4s(2h, k), if h is odd,
s4(h, k) = −4s(h, k) + 8s(h,2k), if h+k is even,
s5(h, k) =−10s(h, k) + 4s(2h, k) + 4s(h,2k)
and each one of S(h, k) (h+k even), s1(h, k) (h odd), s2(h, k) (k odd), s3(h, k) (k even), s4(h, k) (h even) and s5(h, k) (h+k odd) is zero.
Recently, relations between Hardy-Berndt sums and theta function were studied cf. ([6], [7], [29], [23], [24], [26], [36]).
The manin motivations of this paper are given as follows:
In Section 2, we give some identities related to the Weierstrass ℘(z)- function, Hardy-Bernd sums, theta functions and generalized Dedekind eta function. We give relations between the Weierstrass ℘(z)-function, Hardy- Bernd sums, theta functions and generalized Dedekind eta function.
In Section 3, we define generalized Hardy-Bernd sums. By using these generalizations, we construct elliptic analogue of the generalized Hardy- Berndt sums related to elliptic Bernoulli polynomials.
2 Generalized Dedekind Eta function
In this section, we give relations between the Weierstrass℘-function, Dedekind sums, Hardy-Berndt sums and generalized Dedekind eta functions. In [37], Tzeng and Miao defined the following relations related to the generalized Dedekind eta function:
η2(2τ) = 1
2η01(τ,2)η2(τ), η2
τ + 1 2
=η10(τ,2)η2(τ), η2
τ+ 1 2
=eπi12η11(τ,2)η2(τ), η(3τ) = 1
√3η10(τ,3)η(τ), η
τ 3
=η10(τ,3)η(τ),
η
τ+ 1 3
=eπi36η11(τ,3)η(τ), η
τ+ 2 3
=eπi18η11(τ,3)η(τ). By the above equations and (2), we have
(5) ϑ3(z) = η2(z)
η10(z)η(z+ 1), cf.[20], and
logϑ3(z) = logη(z)−logη10(z)− πi
12, cf.[20].
By using (5), we obtain
(6) logη10(Az) = logη(Az)−logϑ3(Az)− πi 12, where A∈Γ(1).
The Weierstrass ℘-function is defined as follows:
Let Λτ =Z+τZ, τ ∈H be a lattice and z ∈C
℘(z; Λτ) = 1
z2 +
0=w∈Λτ
1
(z−w)2 − 1 w2
cf. ([39], [22], [38]).
Relation between the function ℘ and ϑ3 is given by
(7) y(z) =℘(1
2)−℘(z
2) = π2ϑ43(z) cf. ([15], [24]).
By using (6) and (7), we have the following theorem:
Theorem 3.
(8) logη10(Az) = logη(Az)− logy(Az)−2 logπ
4 − πi
12
By using(8), we obtain logη10(Az
2) = logη(Az
2)−logy(Az2)−2 logπ
4 − πi
12 Substitutingz = τ−cd into Theorem 1, we have
(9) logη(z
2) = logη(a− 1τ
2c )− πi(a+d−6c)
24c +πis(d,2c)− 1
2log(τ), and
(10) logη(2z) = logη(2a− 21τ
c )−πi(2a+ 2d−3c)
12c +πis(2d, c)−1
2log(τ).
By substituting (9) and (10) into (2), if c+d is odd, then we obtain (11) logϑ3(Aτ) = logϑ3(τ) + πi
4(S(d, c)−4)− 1 2logτ, where S(d, c) denotes Hardy-Berndt sum.
By using (7), (8) and (11), we arrive at the following theorem:
Theorem 4. If a+d is odd, then we have
logη10(Aτ) = logη10(τ)−πiS(d, c) + 4s(d, c)−3
4 + logτ.
If a+d is even, then we have
logη10(Aτ) = logη10(τ)−πis5(d, c) + 2s(d, c)
2 − 3πi
4 + logτ.
3 Elliptic analogue of generalized Hardy-Berndt sums
Recently, elliptic Bernoulli polynomials, elliptic analogue of the generalized Dedekind-Rademacher, Dedekind-Apostol and Hardy-Berndt sums have stud- ied by many mathematicians (for detail see [10], [6], [7], [9], [13], [17], [18], [21], [22], [25], [26], [12], [5])
In this section, we introduce generalized Hardy-Berndt sums and elliptic analogue of the generalized Hardy-Berndt sums related to elliptic Bernoulli polynomials.
Hardy-Berndt sums in (4), are redefined as follows:
Leth and k be integers withk > 0,the Hardy-Berndt sums are defined as follows
S(h, k) = 4
k−1
j=1
(((h+k)j 2k )), (12)
s1(h, k) = k
j=1
((j k))
2((hj
k ))−4((hj 2k))
,
s2(h, k) =−4
k−1
j=1
((hj 2k)), s5(h, k) =
k j=1
((j k))
2((hj
k ))−4(((h+k)j 2k ))
,
for detail see cf. ([8], [32], [27], [24]). By using (12) and Bernoulli functions, we arrive at the fallowing definition ([8], [27]):
Definition 2. Let h and k be integers with k > 0, the Hardy sums are defined as follows:
S(h, k :m) = 4
k−1
j=1
Bm((h+k)j 2k ), s1(h, k :m) =
k−1
j=1
B1(j
k)(2Bm(hj
k )−4Bm(hj 2k))
= 2s(h, k:m)−4
k−1
j=1
B1(j
k)Bm(hj 2k),
s2(h, k :m) =
k−1
j=1
(−1)jB1(j
k)Bm(hj k ), s3(h, k :m) = −4
k−1
j=1
(−1)jBm(hj k ),
s4(h, k:m) = −4
k−1
j=1
Bm(hj k ), s5(h, k:m) =
k−1
j=1
B1(j k)
2Bm(hj
k )−4Bm(jh+k 2k )
= 2s(h, k:m)−4
k−1
j=1
B1(j
k)Bm((h+k)j 2k ), where s(h, k :m) is generalized Dedekind sums, which is defined by
s(h, k :m) =
jmodk
j
kBm(hj
k ) cf. ([1], [3], [2], [4], [32], [30], [25]).
By using Definition 2 and (1), we define (13) S2,m,n(a, b, c:x, y, z) =
kmodc
(−1)kBm(ak+z
c −x)Bn(bk+z c −y).
By using (3) and (13), we arrive at the following Theorem:
Theorem 5. Let a, a, b, b, c, c be positive integers and x, x, y, y, z, z be real numbers. Suppose that
az −cx ∈/< a, c >Z and bz −cy ∈/< b, c >Z,
where < a, b > is the greatest common divisor of a and b.
S2τ,m,n(−→a ,−→
b ,−→c;−→x ,−→y ,−→z) = jmodc lmodc
(−1)j+lBm
al+z
c −x,aj+z
c −x;a aτ
×Bn
bl+z
c −y,bj+z c −y;b
bτ
. Remark 1. If m= 1, n= 1, or if az−cx /∈< a, c >Z and bz−cy /∈< b, c > Z, then
τlim→i∞S2τ,m,n(−→a ,−→
b ,−→c;−→x ,−→y ,−→z ) =S2,m,n(a, b, c;x, y, z), Let a, b and c be positive integers, and x, y and z be real numbers.
S2,m,n(a, b, c:x, y, z) =
kmodc
(−1)kBm(ak+z
c −x)Bn(bk+z c −y).
If m = a = 1 and x = y = z = 0, then S2,m,n(a, b, c : x, y, z) reduces to s2(b, c:m).
By using Definition 2 and (1), we define S5,m,n(a, b, c : x, y, z) = 2
kmodc
Bm(ak+z
c −x)Bn(bk+z c −y)
−4
kmodc
Bm(ak+z
c −x)Bn((b+c)k+z 2c −y), or
(14) S5,m,n(a, b, c:x, y, z) = 2Sm,n(a, b, c:x, y, z)−4Y5,m,n(a, b, c:x, y, z), where Sm,n(a, b, c : x, y, z) denotes an analogue of generalized Dedekind- Rademacher sums and
Y5,m,n(a, b, c:x, y, z) =
kmodc
Bm(ak+z
c −x)Bn((b+c)k+z 2c −y).
By using (3) and (14), we arrive at the following Theorem:
Theorem 6. Let a, a, b, b, c, c be positive integers and x, x, y, y, z, z be real numbers. Suppose that
az −cx ∈/< a, c >Z and bz −cy ∈/< b, c >Z, where < a, b > is the greatest common divisor of a and b.
S5τ,m,n(−→a ,−→
b ,−→c;−→x ,−→y ,−→z ) = 2 jmodc lmodc
Bm
al+z
c −x,aj+z
c −x;a aτ
×Bn
bl+vz
c −y,bj+z c −y;b
bτ
−4 jmodc lmodc
Bm
al+z
c −x,aj+z
c −x;a aτ
×Bn
bl+z
c −y,(b+c)j+z 2c −y;b
bτ
. Remark 2. If m= 1, n = 1, or if az−cx /∈< a, c >Z and
bz−cy /∈< b, c >Z, then
τlim→i∞S5τ,m,n(−→a ,−→
b ,−→c;−→x ,−→y ,−→z ) =S5,m,n(a, b, c;x, y, z).
Let a, b and c be positive integers, and x, y and z be real numbers. If m = a = 1 and x = y = z = 0, then S5,m,n(a, b, c : x, y, z) reduces to s5(b, c : n). Observe that elliptic analogue of the s1(h, k : n) is similar to that of s5(b, c:n).
Now, we define generalized Hardy-Berndt sum’s sj(h, k : n), j = 1,3,4 and S(h, k :n) as follows:
(15) S4,0,n(0, b, c; 0, y, z) =−4
kmodc
Bn(bk+z 2c −y),
(16) S1,m,n(a, b, c:x, y, z) = 2Sm,n(a, b, c:x, y, z)−4Y1,m,n(a, b, c:x, y, z), where Sm,n(a, b, c : x, y, z) denotes an analogue of generalized Dedekind- Rademacher sums and
Y1,m,n(a, b, c:x, y, z) =
kmodc
Bm(ak+z
c −x)Bn(bk+z 2c −y), and
S3,0,n(0, b, c; 0, y, z) = −4
kmodc
(−1)kBn(bk+z c −y), (17)
SH,0,n(0, b, c; 0, y, z) = −4
kmodc
Bn((b+c)k+z 2c −y).
Note that substituting x = y =z = 0 in the above, then Sk,m,n(a, b, c : x, y, z), k = 1,3,4 and SH,0,n(0, b, c; 0, y, z) reduce to sj(h, k :n), j = 1,3,4 and S(h, k :n), respectively.
By using (3) and (15), we construct elliptic analogue of sj(h, k : n), j = 1,3,4 andS(h, k :n) sums by the following theorem:
Theorem 7. Let a, a, b, b, c, c be positive integers and x, x, y, y, z, z be real numbers. Suppose that
az−cx ∈/< a, c >Z and bz −cy ∈/< b, c >Z,
where < a, b > is the greatest common divisor of a and b.
SH,τ0,n(−→ 0,−→
b ,−→c;−→
0,−→y ,−→z ) = jmodc lmodc
Bn
bl+z
c −y,(b+c)j+z 2c −y;b
bτ
,
S1τ,m,n(−→a ,−→
b ,−→c;−→x ,−→y ,−→z) = 2 jmodc lmodc
Bm
al+z
c −x,aj+z c −x;a
aτ
×Bn
bl+z
c −y, bj +z
c −y;b bτ
−2 jmodc lmodc
Bm
al+z
c −x, aj+z c −x;a
aτ
×Bn
bl+z
c −y,bj +z 2c −y;b
bτ
,
S3τ,0,n(−→ 0,−→
b ,−→c;−→
0,−→y ,−→z) = jmodc lmodc
(−1)j+lBn
bl+z
c −y, bj+z c −y;b
bτ
,
S4τ,0,n(−→ 0,−→
b ,−→c;−→
0,−→y ,−→z ) = jmodc lmodc
Bn
bl+z
c −y, bj+z
2c −y;b bτ
.
Remark 3. If m= 1, n = 1, or if az−cx /∈< a, c >Z and
bz−cy /∈< b, c > Z, then
τlim→i∞Sk,τ0,n(−→ 0,−→
b ,−→c;−→
0,−→y ,−→z) = Sk,0,n(0, b, c; 0, y, z), k= 3,4,
τlim→i∞SH,τ 0,n(−→ 0,−→
b ,−→c;−→
0,−→y ,−→z) = SH,0,n(0, b, c; 0, y, z),
τlim→i∞S1τ,m,n(−→a ,−→
b ,−→c;−→x ,−→y ,−→z) = S1,m,n(a, b, c;x, y, z).
Acknowledgement 1 This paper was supported by the Scientific Research Project Administration of Akdeniz University.
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Department of Mathematics Faculty of Science
University of Akdeniz 07058 Antalya, Turkey
Email addresses: yilmazsimsek@hotmail.com
